Differentiation with the quotient rule

Question

I have the question:

Given that $$y=\frac{e^x}{\sqrt{1+2x}}$$

Show that $$\frac{dy}{dx} = \frac{2xe^x}{\sqrt{(1+2x)^3}}$$

I've done the question but I got $2xe^x(\sqrt{(1+2x)^3})$. I feel like this is too similar to the sheet's answer, so am I wrong or is the sheet printed wrong?

Working:

Let $u = e^x$, $v = 1+2x$, $w = \sqrt{v}$.

Using the chain rule, $$\frac{dw}{dx} = \frac{dw}{dv} * \frac{dv}{dx}$$

So $$\frac{dw}{dx} = 2(\frac{1}{2}v^{-\frac{1}{2}}) = \frac{1}{\sqrt{v}} = \frac{1}{\sqrt{1+2x}}$$

And $\frac{du}{dx}$ is clearly $e^x$.

Therefore, using the quotient rule, $$\frac{dy}{dx} = \frac{e^x(\sqrt{1+2x})- e^x(\frac{1}{\sqrt{1+2x}})}{(1+2x)^2}$$

$$=\frac{e^x(\sqrt{1+2x}-\frac{1}{\sqrt{1+2x}})}{(1+2x)^2}$$

$$=\frac{e^x(\frac{2x}{\sqrt{1+2x}})}{(1+2x)^2}$$

$$=\frac{\frac{2xe^x}{\sqrt{1+2x}}}{(1+2x)^2}$$

Using the rule

$$\frac{x^m}{x^n} = x^{m-n}$$

We get

$$\frac{2xe^x}{(1+2x)^{-\frac{3}{2}}}$$

$$=2xe^x(\sqrt{1+2x})^3 $$

Have I done this correctly?

Answer 1

I suspect

$$\frac{1}{\left(\sqrt{1+2x}\right)^2} \not = \frac{1}{\left({1+2x}\right)^2}$$

and $$\frac{\frac{1}{\sqrt{1+2x}}}{(1+2x)^2} \not =\frac{1}{(1+2x)^{-\frac{3}{2}}}$$

but $$\frac{\frac{1}{\sqrt{1+2x}}}{\left(\sqrt{1+2x}\right)^2} = \frac{1}{\left(\sqrt{1+2x}\right)^3}$$

Answer 2

The error is in $$\frac1{\sqrt{2x+1}} \cdot \frac1{(2x+1)^2} = \frac1{(2x+1)^{2+\frac12}} = \frac1{(2x+1)^{\frac52}}$$ and $$\frac1{\sqrt{2x+1}^2} \ne \frac1{(2x+1)^2}$$ so there is an error in the quotient rule. (the exponent is positive in the denominator, not negative)
Tracing the errors through the equation we obtain $$\frac1{\sqrt{2x+1}} \cdot \frac1{(\sqrt{2x+1})^2} = \frac1{\sqrt{2x+1}^3} = \frac1{(2x+1)^{\frac32}}$$

Answer 3

Using Quotient rule: $$\frac{dy}{dx} = \frac{e^x(\sqrt{1+2x})-e^x(\frac{1}{\sqrt{1+2x}})}{1+2x}.$$ You can simplify now.